A rate of convergence for loop-erased random walk to SLE(2) Michael J. Kozdron University of Regina http://stat.math.uregina.ca/ ∼ kozdron/ Conformal maps from probability to physics Centro Stefano Franscini May 24, 2010 Based on joint work with Christian Beneˇ s (Brooklyn College, CUNY) and Fredrik Johansson Viklund (KTH, soon Columbia)
Introduction The Schramm-Loewner evolution with parameter κ (SLE κ ) was introduced in 1999 by Oded Schramm while considering possible scaling limits of loop-erased random walk. Since then, it has successfully been used to study various other lattice models from two-dimensional statistical mechanics including percolation, uniform spanning trees, self-avoiding walk, and the Ising model. Crudely, one defines a discrete interface on the 1 /N -scale lattice and then lets N → ∞ . The limiting continuous “interface” is an SLE. In “Conformal invariance of planar loop-erased random walks and uniform spanning trees” (AOP 2004), Lawler, Schramm, and Werner showed that the scaling limit of loop-erased random walk is SLE with parameter κ = 2 . The proof is qualitative and no rate of convergence immediately follows from it. Schramm’s ICM 2006 Problem 3.1: “Obtain reasonable estimates for the speed of convergence of the discrete processes which are known to converge to SLE.” 1
b b Review of Radial SLE g t γ ([0 , t ]) W t Reparametrize γ so that t (0) = e t . g ′ This is the capacity parametrization. 2
Review of Radial SLE (cont) The evolution of the curve γ ( t ) , or more precisely, the evolution of the conformal transformations g t : D t → D , can be described by the Loewner equation. For z ∈ D with z �∈ γ [0 , ∞ ] , the conformal transformations { g t ( z ) , t ≥ 0 } satisfy ∂t g t ( z ) = g t ( z ) W t + g t ( z ) ∂ W t − g t ( z ) , g 0 ( z ) = z, where W t = z → γ ( t ) g t ( z ) . lim We call W the driving function of the curve γ . The radial Schramm-Loewner evolution with parameter κ ≥ 0 with the standard parametrization (or simply SLE κ ) is the random collection of conformal maps { g t , t ≥ 0 } obtained by solving the initial value problem ∂t g t ( z ) = g t ( z ) e i √ κB t + g t ( z ) ∂ , g 0 ( z ) = z. (LE) e i √ κB t − g t ( z ) where B t is a standard one-dimensional Brownian motion. 3
From LERW to SLE n Z 2 grid domain • Let D ∋ 0 be a simply connected planar domain with 1 approximation D n ⊂ C . A grid domain is a domain whose boundary is a union of edges of the scaled lattice. That is, D n is the connected component containing 0 in the complement of the closed faces of n − 1 Z 2 intersecting ∂D . Note that D n is simply connected. • ψ D n : D n → D , ψ D n (0) = 0 , ψ ′ D n (0) > 0 . • γ n : time-reversed LERW from 0 to ∂D n (on 1 n Z 2 ). • ˜ γ n = ψ D n ( γ n ) is a path in D . Parameterize by capacity. • W n ( t ) = W 0 e iϑ n ( t ) : the Loewner driving function for ˜ γ n . 4
L-S-W Prove Convergence of the Driving Processes Theorem (Lawler-Schramm-Werner, 2004). Let D be the set of simply connected grid domains with 0 ∈ D, D � = C . For every T > 0 , ε > 0 , there exists n = n ( T, ε ) such that if D ∈ D with inrad ( D ) > n , then there exists a coupling between loop-erased random walk γ from ∂D to 0 in D and Brownian motion B started uniformly on [0 , 2 π ] such that � � sup | θ ( t ) − B (2 t ) | > ε < ε, P 0 ≤ t ≤ T where θ ( t ) satisfies W ( t ) = W (0) e iθ ( t ) and W ( t ) is the driving process of γ in Loewner’s equation. This is “a kind of of convergence” of LERW to SLE 2 , and leads (without too much difficulty) to the stronger convergence (which we won’t discuss) of paths with respect to the Hausdorff metric. L-S-W then use this result to establish convergence of paths with respect to the metric that identifies curves modulo reparametrization (c.f., Aizenman-Burchard). 5
Statement of Main Result Our main result provides a rate for the convergence of the driving processes. Theorem (Beneˇ s-Johansson-K, 2009). Let 0 < ǫ < 1 / 36 be fixed, and let D be a simply connected domain with inrad ( D ) = 1 . For every T > 0 there exists an n 0 < ∞ depending only on T such that whenever n > n 0 there is a coupling of γ n with Brownian motion B ( t ) , t ≥ 0 , where e iB (0) is uniformly distributed on the unit circle, with the property that � � | W n ( t ) − e iB (2 t ) | > n − (1 / 36 − ǫ ) < n − (1 / 36 − ǫ ) . sup P 0 ≤ t ≤ T Recall that W n ( t ) = W n (0) e iϑ n ( t ) , t ≥ 0 , denotes the Loewner driving function for the curve ˜ γ n = ψ D n ( γ n ) parameterized by capacity. 6
Ideas of Proof L-S-W follow the “three main steps” to proving convergence of the driving processes. 1. Find a discrete martingale observable for the LERW path. Prove that it converges to something conformally invariant. 2. Use Step 1 together with the Loewner equation to show that the Loewner driving function for the LERW is almost a martingale with “correct” (conditional) variance. 3. Use Step 2 and Skorokhod embedding to couple the Loewner driving function for the LERW with a Brownian motion and show that they are uniformly close with high probability. To obtain a rate we have to re-examine the steps to find explicit bounds on error terms. 7
Some Notation D � = C is a simply connected grid domain containing the origin V = V ( D ) is the set of vertices contained in D ψ D : D → D with ψ D (0) = 0 , ψ ′ D (0) > 0 If D is a simply connected domain with a Jordan boundary, it is well-known that ψ D can be extended continuously to the boundary so that if u ∈ ∂D , then ψ D ( u ) = e iθ D ( u ) ∈ ∂ D . Our grid domains, however, will be “Jordan minus a slit.” This means that a boundary point of D may correspond under conformal mapping to several points on the boundary of the unit disk. To avoid using prime ends, we adopt the L-S-W convention of viewing the boundary of Z 2 ∩ D as pairs ( u, e ) of a point u ∈ ∂D ∩ Z 2 and an incident edge e . We write V ∂ ( D ) for the set of such pairs, and if v ∈ V ∂ ( D ) , then the notation ψ D ( v ) means lim z → u ψ D ( z ) along e , and this limit always exists. 8
Step 1. A Martingale Observable The martingale observable used by Lawler-Schramm-Werner is the discrete Poisson kernel. Fix n and z ∈ V ( D n ) and let M k = M k ( z ) := H k ( z, γ n ( k )) H k (0 , γ n ( k )) , k ≥ 0 . If x ∈ D n \ γ n [0 , k ] , then H k ( x, γ n ( k )) denotes the probability that simple random walk started at x exits the slit domain D n \ γ n [0 , k ] at γ n ( k ) , i.e., discrete harmonic measure. One can show that M k is a martingale with respect to γ n [0 , k ] , and for fixed k , M k ( z ) is discrete harmonic. 9
Discrete and Continuous Poisson Kernel The next step is to show that for appropriate z when n is large, the discrete and continuous Poisson kernel are close: 1 − | ψ k ( z ) | 2 H k ( z, γ n ( k )) H k (0 , γ n ( k )) ≈ | ψ k ( z ) − ψ k ( γ n ( k )) | 2 with explicit error terms (in terms of the lattice scale 1 /n ). Here ψ k : D n \ γ n [0 , k ] → D . K-Lawler, 2005, did a similar estimate, but worked in a slightly different setting (union of squares domains) with simply connected Jordan domains only. The goal then was different and that result was not optimal for that setting. Those ideas, however, can be modified to apply to the present setting, and optimality can be attempted. (K-L studied two arbitrary points not too close to each other; B-J-K need one “interior” point and one point “next to the boundary.”) 10
Discrete and Continuous Poisson Kernel Theorem (Beneˇ s-Johansson-K, 2009). Let 0 < ǫ < 1 / 6 and let 0 < ρ < 1 be fixed. Suppose that D is a grid domain satisfying n ≤ inrad ( D ) ≤ 2 n . Furthermore, suppose that x ∈ D ∩ Z 2 with | ψ D ( x ) | ≤ ρ and u ∈ V ∂ ( D ) . If both x and u are accessible by a simple random walk starting from 0 , then 1 − | ψ D ( x ) | 2 H D ( x, u ) | ψ D ( x ) − ψ D ( u ) | 2 · [ 1 + O ( n − (1 / 6 − ǫ ) ) ] . H D (0 , u ) = Note. The condition | ψ D ( x ) | ≤ ρ ensures that x is an “interior” point. 11
Proving the Closeness of the Poisson Kernels The proof relies on the fact that for all ε > 0 , we can find δ > 0 such that if • D is a(n appropriate) grid domain, • E ⊂ ∂D is a union of edges of Z 2 , • x is far enough from ∂D , then H ( x ) ≥ ε ⇒ h ( x ) ≥ δ, ( ∗ ) where h ( x ) = h D ( x, E ) is discrete harmonic measure of E from x and H ( x ) is its continuous analogue. The implication in ( ∗ ) is generally not satisfied by grid domains (problems arise with fjords or channels). 12
Proving the Closeness of the Poisson Kernels To get around this problem, one can cut off anything in the domain that isn’t accessible by random walk started at x by creating a Union of Big Squares (UBS) domain D 0 , which is the union of squares of side length 2 centred at the points of V 0 ( D ) , i.e., those vertices in the connected component containing 0. Beurling’s theorem implies that if the Poisson kernels are close in D 0 , they are close in D . Specifically, if n ≤ inrad ( D ) ≤ 2 n , then ∂ψ D ( D 0 ) ⊂ A (1 − cn − 1 / 2 , 1) . The conformal map from ψ ( D 0 ) to D is almost the identity and one can show that ψ D 0 ( x ) = ψ D ( x ) + O ( n − 1 / 2 log n ) and e iθ D 0 ( u ) = e iθ D ( u ) + O ( n − 1 / 4 ) . 13
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